Dynamics of counterpropagating waves in parametrically forced, large aspect ratio, nearly conservative systems with nonzero detuning

Dynamics of counterpropagating waves in parametrically

forced, large aspect ratio, nearly conservative systems with

nonzero detuning

José M. Vega , Edgar Knobloch

E.T.S.I. Aeronáuticos, Universidad Politécnica de Madrid, Plaza Cardenal Cisneros 3, 28040 Madrid, Spain Department of Physics, University of California, Berkeley, CA 94720, USA

Department of Applied Mathematics, University of Leeds, Leeds LS2 9JT, UK

Abstract

The dynamics of parametrically driven, slowly varying counterpropagating wave trains in nearly conservative systems are considered. The system is assumed to be invariant under reflection and translations in one direction, and periodic boundary conditions with period L are imposed, with L large but not too large in order that the eífect of detuning be significant. The dynamics near the minima of the resulting resonance tongues are described by a system of coupled nonlocal Schródinger equations with damping and parametric forcing. Elsewhere the long time behavior of the system is described by a damped complex Duffing equation with real coefficients, whose solutions relax to spatially uniform standing waves. Near the bicritical points where two adjacent resonance tongues intersect a pair of coupled damped complex Duffing equations captures the properties of both puré and mixed modes, and of the periodic solutions resulting from a Hopf bifurcation on the branch of mixed modes. As an application, we consider a Faraday system in an annulus in which a pair of counterpropagating surface gravity-capillary waves are excited parametrically by vertical vibration of the container, including the mean flow driven by time-averaged Reynolds stresses due to oscillatory viscous boundary layers along the bottom and the free surface. This mean flow is shown to have a large eífect near the bicritical point, where the mean flow changes the dynamics of the system both quantitatively and qualitatively. In particular, inclusión of the mean flow permits Hopf bifurcations from the branches of puré standing waves, and parity-breaking bifurcations from mixed modes.

PACS: 47.20.Ky; 47.20.Ma; 47.35,+i; 47.54.+r

1. Introduction

In this paper we consider a pair of coupled, weakly damped, parametrically forced Schrodinger equations that describe the weakly nonlinear evolution of a pair of slowly modulated, counterpropa-gating wavetrains in extended, nearly conservative systems invariant under reflection and translation in one spatial dimensión. Elsewhere (Martel et al., 2000; Higuera et al, 2002b) we have discussed the limit of small detuning, and shown that in this regime the dynamics near onset is described by a pair of coupled nonlocal Schrodinger equations with damping and parametric forcing. These equations describe the dynamics near the minimum of each resonance tongue. However, in systems that are large but not too large the allowed wavenumbers may differ from the wavenumber selected via the dispersión relation and the difference between the response frequency and (half the) forcing frequency may be substantial. In the following we cali this difference the detuning and discuss the dynamics when the detuning is nonzero, Le., the dynamics near the boundaries of the resonance tongues away from their minima.

Fig. 1. Resonance tongues (JI = ±(v — lian)) showing the instability threshold for the fíat state. Owing to the invariance of the amplitude equations under the operation Á^ —> iA^,¡i —> —¡i the diagram for ¡i < 0 is obtained via reflection in

H = 0.

work. Regrettably, there are at present no experimental studies of the excitation of mean flows near such points in large domains (cf. Douady et al, 1989, Fig. 1), although mode interactions in small aspect ratio systems have been studied (Ciliberto and Gollub, 1985; Simonelli and Gollub, 1989).

Against this background, the remainder of the paper is organized as follows. The basic amplitude equations governing the evolution of the complex amplitudes of the two counterpropagating waves are summarized in Section 2, together with a brief summary of the structure of the resonance tongues associated with a parametric instability (Fauve, 1995). The weakly nonlinear dynamics near threshold for generic valúes of detuning is considered in Section 3, while Section 4 focuses on the dynamics near a bicritical point, where neighboring resonance tongues intersect. The Faraday system is considered in Section 5, followed by concluding remarks in Section 6. In each case, we derive the leading order amplitude (or amplitude-mean flow) equations and discuss their properties. It should be emphasized that although in the limit considered in this paper the aspect ratio is large it is not so large that spatial modulations of the surface waves enter into the theory. This is because the theory is developed for forcing amplitudes that are appropriately cióse to threshold. For larger aspect ratios spatial modulations must be taken into account. This may be achieved via mode truncation as in Decent and Craik (1999), or via a systematic asymptotic expansión focusing on the effects of dispersión as in Martel et al. (2003) and references therein. The coupling between these effects and the viscous mean flow in a nearly inviscid Faraday system is explored in Lapuerta et al. (2002). These effects do not enter in the regime studied in the present paper.

2. Amplitude equations and stability of the flat state

In an extended nearly conservative system the amplitudes of slowly varying counterpropagating wavetrains, driven parametrically by external vibration, satisfy equations of the form (Martel et al. (2000), Eqs. (5)-(7))

¿t^4

(2.1)

These equations are valid provided that

\Af

\<\Af\ 4^141, ¡AfKlA+Kl, (2.3)

and the real parameters ¡i (forcing amplitude), v (detuning), and 5 (damping rate) are small. The dispersión coefficient a and the coefficients /?, y of the nonlinear terms are also real but of order unity. In writing these equations we have shifted the phase of A^ by n/4 relative to the equations used in Martel et al. (2000).

These equations are invariant under reflection and translations, represented by the operations

A+ <r+A~ and A± -»• étícA± for all c. (2.4)

In fact Eqs. (2.1) and (2.2) remain invariant (after a rescaling to restore the unit group velocity) under the transformation

v ->• v + 2n/L - a(2n/L)2, A± -»• Q±^ILA±. (2.5)

This transformation allows us to restrict the detuning to a neighborhood of the interval

- n/L < v sí n/L. (2.6) The limit considered in Martel et al. (2000),

§ ~ v ~ n~L~24l, (2.7)

corresponds to (a) a moderately large aspect ratio (so detuning is still felt), (b) a forcing frequency that yields an appropriately small detuning, in addition to (c) an appropriately small threshold ac-celeration to overeóme the (assumed) small dissipation. These conditions restrict the validity of the theory to the vicinity of the minimum of the primary resonance tongue. As shown in Martel et al. (2000) in this regime Eqs. (2.1) and (2.2) reduce to a pair of coupled nonlocal Schrodinger equa-tions. Identical analysis holds near the minima of the other resonance tongues, corresponding to the excitation of modes with modulation wavenumber m) provided |v — 2mn/L\ ~ L~2. However, this

reduction no longer applies if condition (b) is relaxed. It is this case, viz.

\v\~L~\ v ^ 0 , (2.8)

that is of interest here. In the following we therefore focus on the limit:

5L41, | v | - L_ 1< ^ l , \n-v\<\v\, (2.9)

where we are anticipating that the instability threshold is ¡ic ~ v. This limit is the counterpart of

(2.7) (after relaxing the condition |v| 4L~X), except for the fact that we are not making at this stage

any assumption about the order of magnitude of the damping rate 5 except that it is small compared to L~\

To study the limit (2.9) we scale variables and parameters according to

(x,t) = L(x,t), A±=L~1A±, (v, n) = L~\v, /t). (2.10)

Dropping the tildes we obtain (cf. Martel et al., 2000, Eqs. (11) and (12))

Af^Af + i ( v ^ - fiAT) = -5LA± + iL-1[(J3\A±\2+y\AZf\2)A± + aAfx] + ••-, (2.11)

It is these equations that are the subject of Section 3. We emphasize that the properties of these equations differ from those obtained on the basis of fundamentally small aspect ratio theories, even for the primary standing waves. This is a consequence of the scaling (2.10b) which limits the theory that follows to smaller amplitudes than those accessible in small aspect ratio theories. In particular, we do not expect to see secondary saddle-node bifurcations on the standing wave branches, although our results do reproduce the direction of branching of the primary standing waves from such theories (cf. Umeki, 1991). Additional differences, arising from the presence of a large number of modes in the present system, are discussed in Section 3.

The linear stability of the flat solution A^ = 0 of (2.11) and (2.12) is analyzed by considering infinitesimal perturbations of the form exp(Xt) with a wavenumber 2nm. The dispersión relation is

(X + 5Lf + [v - 2nm + a(2nm)2L-1]2 - ¡i2 = 0. (2.13)

Thus the threshold for the instability of the flat state is associated with the mode m = 0 (see Fig. 1), and is given by ¡i = v + S2L2/(2v) + • • • (Le., according to (2.9a) and the rescaling (2.10), ¡i w v, as anticipated above). The remaining (infinitely many) modes are weakly damped (because 5L4,\) and thus cannot be ignored a priori. These are either of the form

A±=A±¿*»"t+2™¿> with A- = A+A+, (2.14)

or of the form

¿±=¿±e±i(-*.<+2-«) with A-= A-A+^A+/A+, (2.15)

where to leading order (as \¡i — v\ <^ |v|, L —>• oo, and 5L —>• 0)

^ , -,1/? , i + ±sm—2nm + v [1 — v/(mn)]1/2 =F 1

sm = [(2nm)2-4nmvf2 and A% = — = y- -( 1/2 ^ , , (2.16)

v [1 - v/(mn)Y/¿ ± 1

for each integer m ^ 0. In (2.15) we are using the identity

A+mA~ = l, (2.17)

which follows from (2.16). Note that

\A+\ < 1 f o r a l l m ^ O if v ^ 0 , and A+ -»• 0 as m ->• oo. (2.18)

3. Dynamics in the generic case: v ^ O, n

In order to understand the nature of the limit (2.9) and the origin of the complexity that must be expected, we consider two cases, depending on the order of magnitude of the damping rate.

3.1. Significant damping: 5 ~ L~3^2

For this case we introduce the damping rate parameter 5, the bifurcation parameter fi, and a slow time variable x defined by

5=L-3/25, n = v + L~1jl, and x=L~1/2t, (3.1)

and seek a solution in terms of the eigenmodes (2.14) and (2.15):

+ Y, e±Ü%mx[(At + L~V2Ami + ' ' -ySmt + (Bt + L~V2Bmi + ' ' Oe"*"']. (3.2)

Here, according to (2.14) and (2.15), the complex amplitudes Á^0, A^ and B^ are such that

^oo=A)o> Bm=AmAm. (3-3)

These and the higher order terms Amj and Bmj depend only on the slow time variable x. Substituting

(3.1) and (3.2) into (2.11) and setting to zero the coefficients of e±1Smt±l27imx at order 0{L~XI2) we

obtain

iv(A± - A*) = -dA+fdx - 5A±, (3.4)

i v ( 4 U £ -B^) = -dAt/dx-5At, (3.5)

HA-B^ - Alx) = -dBt/dx - 5Bt. (3.6)

Eq. (3.3a) now shows that the two equations (3.4) are identical. Thus (3.4) yields no solvability con-dition. However, Eqs. (3.5) and (3.6) do require a solvability concon-dition. Using (2.17) this condition can be written in the form

[l-(A+)2](dAt/dx + 5At) = 0, (3.7)

implying that

A^ü —> 0 exponentially a s z ^ o o . (3-8)

This property of the solution is a consequence of the magnitude of the damping rate selected by (3.1), cf. (2.9). Thus, after a transient, we can take

A± = 0 for all m^O. (3.9)

This fact, together with (3.3a), results at order 0 ( L_ 1) in the pair of equations

The resulting solvability condition can be written in the form

d04+ - A- )/dT + d{Atx ~ A~i) = 2i[/2

+ (P + yMoIV

or, invoking (3.4) and denoting AQ0 = A,

A" + 25A' + 52A = 2v[/2 + (p + y)M|2K (3.12)

In the following we exelude the codimension-two point /? + y = 0 and consider therefore the two cases v(j5 + y) > 0, and v(j5 + y) < 0. In the former Eq. (3.12) possesses solutions that blow up in finite time, an observation consistent with the fact that the primary bifurcation from the flat state to the simplest standing waves (steady states of (3.12)) is subcritical. In contrast if

v(p + y)<0, (3.13)

as assumed henceforth, then all solutions of (3.12) are uniformly bounded as % —>• oo. This result follows from the exact relation:

^-[\A'\2 + (S2 - 2vfi)\A\2 - v(j8 + y)\A\4] = -4S\A'\2, (3.14)

di

obtained on multiplying (3.12) by A and adding the result to its complex conjúgate.

For convenience we write Eq. (3.12) in terms of the amplitude and phase ofA, defined by A=RQ10,

R" + 25R' + (51 - e'2)R = 2v[/¿ + (j8 + y)R2]R, (3.15)

0" + 2(5+R'/R)9r = 0. (3.16)

Integrating (3.16) we obtain

O1 =K{)R-2Q~2k (3.17)

(for some constant K()), which implies that 6' —>• 0 exponentially, as % —>• oo. The apparent breakdown

of this argument as R —>• 0 can be overeóme using an equation obtained by summing the product of (3.12) with A, and its complex conjúgate, and integrating the result: AA' —AA =K()Q~2ÓZ, cf. (3.17).

Thus for large times we may take the phase of A to be constant and, in particular, take A to be real. Eq. (3.12) then reduces to the standard Duffing equation, whose solutions converge to a steady state at large times. Thus, no complexity can be expected in this case.

Note that if

L~l/24d<l, (3.18)

(the first condition being required for the validity of conditions (3.8)) Eq. (3.12) will evolve in a nondissipative manner over a time of order of % ~ b~x as described by the conservative complex

Duffing equation

A" - 2v[fi + (p + y)\A\2]A = 0. (3.19)

The counterpart of (3.14) now yields

\A'\2 - 2vfi\A\2 - v(p + y)\A\4 = constant. (3.20)

A second integral is given by the counterpart of (3.17), namely

for some constant K(), generically nonzero. Thus

R" - K¡R~3 = 2v[/¿ + (p + y)R2]R, (3.22)

and henee only steady states and periodic solutions are to be expected at large times. In fact, Eq. (3.22) can be integrated once, yielding

R'2 - K2 = -K2R~2 + v[2/¿ + (p + y)R2/2]R2, (3.23)

for some constant K\. The allowed time-periodic solutions are now easily computed, but all ultimately decay to steady states for T > ¿ >_ 1.

3.2. Small damping: 5 ~ L~2

The simplification described in the preceding section is a consequence of the assumption that 5 7^ 0 (in fact, ¿>>L-1/2, or equivalently that 3^>L~2), for otherwise (3.8) no longer follows from

(3.7). The situation is more interesting for smaller damping rates. Consequently, in this section we consider the case 5 ~ L~2, which in conjunction with (2.9) is the true counterpart of (2.7). To study

this case we retain the parameter fi defined in (3.1) as the bifurcation parameter, but redefine the damping rate parameter 5 and the slow time variable T:

5=L~25, T=L~lt. (3.24)

In fact, we should use two slow time variables in this limit, namely t ~ Ü¡2 and t ~ L, but this

makes the analysis unnecessarily involved. Of course, because only the slowest time variable is used, some of the equations (Eq. (3.30) below) will depend on the small parameter L~x. The expansions

(3.2) must be replaced by

+ E Q±Ü7imXi(At + ¿- 1^ i + ' ' -ySmt + (Bt +L~lB^ + • • O e " ^ ' ] , (3.25)

where the relations (3.3) still hold. Eqs. (3.5) and (3.6) now become

iv(A+A± -Blx) = -(U±/dT - 5At + i[£fl£ " (2nm)2aAt] + i\A\2[(2p + y)A± + yB*]

-m¿t\

4l +2i/?]T(K

|

+ \Bt\

)AÍ

+ i? E KK*I

+ \

n\

Mt + iA^Bf + AtBf)BH (3.26)

iv(A-B± - Alx) = -dBt/dx - SBt + i[(iAl - (2nm)2aB±] + i\A\2[(2p + y)B± + yA%]

-iP\Bt\2Bt+2i^(\At\2 + \Bt\2)Bt

where A = A^J= A00), as above. The solvability conditions for these equations can be written, using

(3.3b), in the form

+ *VAA±IA„ , ^ ± ^ _ o;, ^ + , ± _i a [ 1 + ( y l+)2] (27 l ? w) 2 ^ ±

2 j ± _ i f t n _ L r / ( + ^ 4 ^ | J ± | 2 J ±

[1 - (A+)2](dA±/dx + 8A±) = 2ifiA+A± - ia[l + (A+)2](2nm)2A

+ i((2jg + y)[l + (A+)2] + 2yA+) \A\2A± - ij8[l + (/1+)4]M±|2^

+ 2 i / ? £ ( [ l + (yl+yl+)2]M±|2 + [(A+f + ( ^ + )2] | ^ |2> 4 ±

+ i y j ] [ ( / l + +A+f\At\2 + (1 + y l + y l + )2| ^ |2] ^ , (3.28)

implying that

[1 - (yl+)2](dM±|2/dT + 25\At\2) = 0. (3.29)

Thus once again \A^ \ —>• 0 for all m ^ O , albeit on the slower timescale % ~ 1. With this simplification the counterpart of Eq. (3.12) becomes

L~\A" + 2¿U' + b2A) = 2v[/2 + (L3 + y ) M |2R (3.30)

and the analysis of Section 3.1 carries over verbatim to the present case. In particular no persistent complex dynamics are expected as t —>• oo. Thus contrary to expectation the 5 ~ L~2 case is

identical to the 5 ~ L- 3/2 case, except for the (considerably longer) timescale for the decay of

complex transients.

4. Dynamics near the bicritical point: |v — n\ <^1

Let us consider now the dynamics near the codimension-two point obtained at v=n. For simplicity, we consider the limit of significant damping. Smaller valúes of the damping ratio only lead to complex transients, as in Section 3.2. We introduce the bifurcation parameters fi and v, defined as

fi = n+L~\fi-2n2a), v = % + L~\v - 2n2a), (4.1)

where the shift —27i2a is included to eliminate the effect of dispersión (see below), and use the

same scaling and time variables as in Section 3.1, namely

5=L-3/2$, T=L~l/2t. (4.2)

Eqs. (2.11) and (2.12) now become

Af^Af + in(A± -A*) = -L~1/2(At + 5A*1) + iL_1[(/2 - 2n2a)AT - (v - 2n2a)A±)]

+ iL-1[(P\A±\2 +y\A^\2)A± + aAfx] + •••, (4.3)

A±(x + l,t)=A±(x,t). (4.4)

The eigenfrequencies and the constants defined in (2.16) are now given by

and the eigenfrequencies therefore coincide in pairs, namely

So=Si=0, S-\=S2=2\fl%, S-2=S-Í = 2\[Í>%,... . (4.6)

Since the marginal modes correspond to m = 0, m = 1 we may write for (3.2)

A± = Af + L~1/2Af + L~lAf + --- + {Bf+ L~1/2Bf + L~xBf + • • .)e±2l7ÜX + HOH, (4.7)

where, as in Section 3.1, we have

A" = 4 , B~ = -B+, (4.8)

and the abbreviation HOH stands for spatial harmonics with wavenumbers 2nm, with m ^ 0,1. As in Section 3 these higher harmonics decay to zero exponentially as % —>• oo and can be ignored. Substitution of (4.7) into (4.3) yields, at orders 0 ( ¿ -1 / 2) and 0(L~V),

in(Af-Áf) = -dAf/dx-5Af, (4.9)

in(-Bf -Bf) = -dB±/dx-5B± (4.10)

and

m{Af -Af) = -dAf/dx - 5Af + i(/2 - 2n2a)Af - i(v - 2n2a)A^

+ ip(\Af\2 +2\B±\2)At + iy[(\Af\2 + \Bf\2)Af +AfB*Bf], (4.11)

in(-Bf -Bf) = -dBf/dx - 5Bf + i(/2 - 2n2a)Bf - i(v + 2n2a)B±

+ ij8(|5±|2 + 2\A±\2)B± + iy[(\B*\2 + \Af\2)Bf + BjAfAf]. (4.12)

Using (4.8) we obtain that Eqs. (4.9) and (4.10) are always solvable. But, as in Section 3.1, Eqs. (4.11) and (4.12) possess a solution only if two solvability conditions hold. These lead to the following evolution equations for A = A^ = A^ and B = B^ = —B^:

A" + 25A' + 52A = 2%\¡i - v + p(\A\2 + 2|5|2) + y\A\2]A, (4.13)

B" + 25B' + 52B = 2%\¡i + v - £(|5|2 + 2|^|2) - y|5|2]5. (4.14)

Note that these equations are invariant under the operations

A^y-A, A^emA for all d , B -»• -B, and B -»• e1C25 for all c2, (4.15)

which genérate the group 0 ( 2 ) x 0 ( 2 ) , much larger than the original orthogonal group 0(2), see (2.4). Thus some care must be taken in the interpretation of the results. The additional (and spurious) sym-metry is not broken by higher order terms in (4.3) and (4.4) because the original spatially resonant terms corresponding to wavenumbers N and N+í, viz. ÁNBN and BN~1AN+1, become exponentially

small as N —>• oo and henee are absent from Eq. (2.1). The absence of these terms is important since if present they would split the mixed mode branch into two, characterized by (N +1)<¡>A —N(¡>B = 0,TI,

where <¡>A,B are the phases ofA and B. In addition, there is the possibility of a transition between these two branches via a tertiary branch of traveling waves created at either end through parity-breaking bifurcations (Crawford et a l , 1990). None of this behavior will be present here.

4.1. Steady states and their linear stability

For the sake of simplicity, we assume that

£ + y > 0 . (4.16)

The case /? + y < 0 yields similar results, since Eqs. (4.13) and (4.14) are invariant under the operation

A^B, v - > - v , P^-P, y - » - ? . (4.17)

The puré mode solutions of Eqs. (4.13) and (4.14) are given by

, ,? v — fi + o21(2%) , ,, , , , ? , ?, v + fi — d2/(2n)

As2 = —n - — - , B2=0, and \AS\2 = 0, B2 = —n - — - (4.18)

and correspond to standing waves with adjacent wavenumbers, while the mixed modes are given by

,. , 9 , 9 , , (271/2 — S2,— 271/2 + <r) (v,v)

(i^i

2

-i<i)= w-y)

+

ikrñ

<419)

and correspond to spatially modulated standing waves. In the following we exelude the codimension-two points P — y = 0 and 3/? + y = 0, and take fi as the bifurcation parameter. It follows that the two types of puré modes in (4.18) only exist if fi < /2<j~ and fi > fi^, respectively, where

fif = 52l(2n) ± v (4.20)

are the bifurcation points from the flat state, and that they bifúrcate in opposite directions. Mixed modes only exist if

v/(3j8 + y) > 0 (4.21)

and min{fi^~, fi^} < fi < max{fi^~,fi^}, where

fif = S2/(2n) ±(P- y)v/(3jff + y) (4.22)

(a)

(c)

Po

i \A\2 + \B\2

Y v

K¿-tlH

/¿O

* \A\2 + \B\2

(e)

(b) _{k M | 2 _ L | R | 2}

\¿? + \B? ,

(d)

u+ N

-^ o A*

• |^|2 + |S|2

* |¿|2 + |5|2

m

/¿o

H V>

Fig. 2. Sketch of the stable (—) and unstable ( ) puré and mixed modes of (4.13) for ¡3 + y > 0 and: (a) v > 0,

3p + y > 0, p - y > 0, (b) v > 0, 3p + y > 0, P - y < 0, (c) v > 0, 3p + y < 0, (d) v < 0, 3p + y < 0 (thus P - y < 0\

and (e) v < 0, 3/? + y > 0. Results for /? + y < 0 follow using the symmetry (4.17).

algebraic equations

'a

+

sf

2%

fi + v-2(p + y)\A

\

-2p\B

X! - (p + 7)|A|

X

- 2/?|5

|

(T

+ Y

) = 0,

-{$ + y)\As\2Xx +

2t6\As\2(X1+X2) +

(X + 5? 2%

(X + 5f 2%

fi + v-2(fi + y)\As\2-2p\Bs

fi-v + 2(p + y)\Bs\2+2p\As

X2-2p\Bs\2(Y1+Y2) = 0,

(4.24)

Y1+(t6 + y)\Bs\2Y2 = 0,

(4.25)

2p\As\2(X1 + X2) + (p + y)\Bs\2Yx + (A + 5?

2% fi-v+2(p + y)\Bs\2 + 2p\As Y2 = 0,

yield the characteristic equation (X2 + 2bXf

4n2

'X2 + 2¿A

271 203

+

U2+ 2 ¿ A

271 + 2(i5 + y ) | 5í|2 + 16ñ¿s\2\B3

(4.26)

0,

(4.27)

obtained with the help of (4.19). This equation possesses a double zero eigenvalue, with eigenfunc-tions

(XuX2,YuY2) = (iAs,-iAs,0,0) and (XUX2,YUY2) = (0,0,iAs,-iAs), (4.28)

which result from the invariance of (4.13) and (4.14) under the actions (4.15). The remaining eigenvalues arising from the first factor are stable. Thus we only need to consider the roots of the second factor:

(i) If P — y 7^ 0, 3f¡ + y ^ 0 this factor yields zero eigenvalues only at the bifurcation points corresponding to the appearance of the branch of mixed modes.

(ii) However, purely imaginary eigenvalues, of the form X = ± ü / , are possible provided that XJ/2 = n(P + y X I ^ I2 - \AS\2) = -52 + [54 + 4 ^ ( 3 0 + y)(fi - y)\As\2\Bs\2]ll2 > 0. (4.29)

Since, according to (4.19),

2[/2 - 52l(2n)} .A.2.na v2 [jl-52/(2n)]2

\B, \AS \M\BS (4.30)

¡1-y ' ^ ' '"J | ( 3 ^ + y)2 ($-yf '

we conclude immediately that the corresponding secondary Hopf bifurcation does not occur if ( 3 £ + y ) 0 3 - y ) < 0. If, however, ( 3 £ + y ) 0 3 - y ) > 0 and (4.21) holds, a unique Hopf bifurcation point fi = fÍH is present. The uniqueness of this point follows from an examination of the two quantities appearing in Eq. (4.29) as a function of M = [fí — 32/(2n)]/(t8 — y) in the permitted

range between ±v/(3/? + y).

\o(cl

)

1 bulk

/

////////////////////////

y///////////

0{C

J

)

Fig. 3. Sketch of the fluid domain.

is only a single branch of mixed modes, and that this branch has two zero eigenvalues instead of the expected single zero eigenvalue. In particular, since there are no other eigenvalues in the phase direction, there are no parity-breaking bifurcations from the mixed mode branch. However, the Hopf bifurcation from the mixed modes does not require the presence of the resonant terms, since it occurs in the standing wave invariant subspace.

5. Application to the Faraday system

As mentioned in Section 1, Eqs. (2.1) and (2.2) describe Faraday waves in large aspect ratio, two-dimensional containers, provided that the effect of the mean flow produced by the waves is included. The proper form of this mean flow can be obtained from the general coupled amplitude mean flow (GCAMF) equations, derived in Vega et al. (2001). As shown below, the mean flow has only a minor effect in the generic case considered in Section 3, but its effect near the bicritical point considered in Section 4 is much more dramatic.

5.7. Coupled amplitude-mean flow equations

We consider a two-dimensional fluid layer above a horizontal píate that is vibrated vertically with an appropriately small amplitude (Fig. 3). The layer is laterally unbounded, with periodic boundary conditions. We use a Cartesian coordinate system with the x-axis along the unperturbed free surface and y vertically upwards, and nondimensionalize space and time with the unperturbed depth h and the gravity-capillary time [g/h + T/iph3)]-1^2, where g is the gravitational acceleration, p is the

density and T is the coefficient of surface tensión. The nondimensional equations governing the system then are (Vega et al, 2001)

</fct + Ílyy = £2, &t ~ ¡l/y&x + ¡k&y = CgiQja + Qyy), (5.1)

(i - s)f

- sifj^TTf!^ - i>

+ Atf

-(ik + bf

)a + (4

+ ^

)x/2

+ ( 42 + tá)yfx/2 ~ 4n<±>2 sm(2wt)fx = ~Cg[3\kxy + \\lyyy - (t/xxx + Yxyy)fx\

+2Cg

ry !yJ xl ** -r^vj Miiyx-wiy_( x K-,gl-"Kxy i Yyyy \Yxxx i Va}»}»

íxy ~~ YyyyJJ x ~ Yxyyk*- ~ J x

2rxyñ + (Ax-^yy)fx] , 0 ^ (V5fcc>> ~ V W / x ~ Txyy{ 1 ~ / * ) / ;

1+f:

+ 2Cg yWy vyyyux ^ rxyy,~ JX)Jx ^ y = f,

(5.3) l + / x2

Qydx = \l/ = \l/y = Q aty = -l, (5.4)

tKx + Z,j>,0 = «K*,J>,0, f(x + L,t) = f(x,t), / / d x = 0. (5.5)

Here \¡/ is the streamfünction, such that the velocity (u,v) = (—\J/y,4x), Q is the vorticity, and /

is the free surface elevation required to satisfy volume conservation recalled in (5.5c). Condition (5.4a) is necessary in order that the pressure be periodic in x. The remaining equations and boundary conditions are standard. The resulting problem depends on the aspect ratio L, the nondimensional vibration amplitude ¡i and frequency 2a>, the capillary-gravity number Cg = v/[gh3 + Th/p]1^2, and

the gravity-capillary balance parameter S = T/(T + pgh2), where v is the kinematic viscosity. Note

that Cg and S are related to the usual Onhesorge number C = vlp/iTh)]1^2 and the Bond number B = pgh2/T by Cg = C/(l + 5)1 / 2 and 5 = 1/(1+ B). Thus 0 < S < 1, and the extreme valúes,

5 = 0 and 1, correspond to the purely gravitational (T = 0) and the purely capillary (g = 0) limits, respectively.

In a laterally unbounded layer the driving frequency 2a> selects a wavenumber k near ko, given by the inviscid dispersión relation for surface gravity-capillary waves

co = [(1 - S + Sk¡)h tanh£0]1/2- (5.6)

The associated eigenfunction is proportional to (\J/,f) = (fo, 1), with

f0 = cosmh[ko(y + l)]/(£0sinMo). (5.7)

The GCAMF equations are derived under the following assumptions:

(l+h)(Cg/(o)1/2<l, L>\, |^x| + | ^ | ^ 1 , \f\<\, (5.8)

and the assumption that the spatial Fourier transforms of \¡/ and / both peak for all time around the wavenumbers ±mko, with m = 0,í,... . These require, in particular, that

Cg<\. (5.9)

In a laterally bounded container the wavenumber ko may not fit, and instead the vertical vibration of the container will select wavenumbers near the slightly shifted wavenumber k, defined by

k = 2nN/L, (5.10)

where N > 1 is an integer such that

Thus

\k-ko\ - i_ 1< ^ l . (5.12)

With the wavenumber k the inviscid problem can be embedded in one with periodic boundary conditions. Thus in the following we use k instead of ko as the basic wavenumber.

The above assumptions permit us to decompose the streamfiínction (and vorticity) in the bulk and the free surface elevation into three parts, namely, (i) two parametrically excited counter-propagating wavetrains with frequency co and wavenumbers ±k associated with the surface gravity-capillary modes, modulated slowly in both space and time, (ii) a mean flow depending weakly on time but in principie strongly on the spatial variables x and y, and (iii) the remaining part of the solution, which will be called nonresonant. As discussed in Vega et al. (2001) our assumptions guarantee that the mean flow variables exhibit well-defined averages in the fast variable x (see (5.24) below). Under these conditions the free-surface deflection and the streamfiínction in the bulk can be written in the form

/ = emt(A+elkx +A~Q-lkx) + ce. + HOT + fm +NRT, (5.13)

xj, = W()Qmt(A+Qlkx - A~Q-lkx) + ce. + HOT + ij/m+ NRT. (5.14)

Here the superscript m denotes the mean flow variables, and NRT and HOT stand for nonresonant terms and higher order terms, respectively. The function f 0, defined in (5.7), is evaluated at the

shifted wavenumber k. The complex amplitudes A^ depend weakly on both t and x, while fm, \J/m

and Qm depend weakly on t but strongly on x (and y),

\Af\ + \Af\<\A±\<\, \f?\<\r\<i, m<\r\<h (5.15)

cf (2.3).

The complex amplitudes A^ and the mean flow variables \J/m and fm evolve on a timescale that is

large compared to the basic period 2n/a> according to evolution equations obtained from appropriate solvability conditions (Vega et al, 2001). To adapt the resulting GCAMF equations to the notation and scaling used in (2.1), we must redefine t, ^±, \J/m, and ¡i as

A±=A±/Jv~g, xjjm = xjjmlvg, fí = ncüktanhk/vg, (5.16)

where vg = co'(k) is the group velocity of the surface waves. Dropping tildes, the amplitude equations

become

Af^Af = icüAt -{o + ivM* + m^\

+ yM

|

M

+ itiA

±iTi J g(y)(^)

dyA

+ ir

(f

)

A

, (5.17)

while the equation for the mean flow in the bulk takes the form

C + «/& = Í2m, Í2f - [^ + (M+|2 - \A-\2)g(y)]Q™ + ^Qmy = ( Q K ) ( Í 2 £ + Qmyy\ (5.18)

in — 1 < y < 0, with the boundary conditions

(1 - S)f: - Sflx - v]^t + Cgvg(^yy + 3 « , ) = -r5vg(\A+\2 + \A~\2)X aty = 0, (5.20)

Q™dx = xpm = 0, xJj™ = -r6[iA+A-e¿lkx + c.c. + \A-\¿-\A+\¿] at y =-l, (5.21)

4±r 4±r

A±(x+ L, t)=A ^ , 0 , x¡Jm(x+L,y,t) = x¡jm(x,y,t), fm(x + L,t) = fm(x,t),

fm(x,t) dx = 0.

(5.22)

(5.23)

The latter are derived from a carefül matching between the nonlinear flow in the oscillatory boundary layers (whose thickness is 0 ( Q )) and the flow in the bulk (Vega et al, 2001). The horizontal mean valué (•)* is defined as

{G(x,y,t))x = (2f)-X i G(z,y,t)dz, with M<f<^Z, (5.24)

(5.25) while 5 > 0 and v are given by

5 = (axClJ2 + «2Cg)/vg, v = [2nN/L -k{)- vgaxCf]/vg,

where

a! = £(co/2)1/2/sinh(2£), a2 = [2 + (1 + tanh2£)/(4 sinh2£)]£2.

Finally, the dispersión coefficient a is given by

a = -co"(k)/(2vg).

(5.26)

(5.27)

The remaining coefficients in (5.17), (5.19)—(5.21), fi,y,r\,. ..,r$, and the function g have also been computed, and are given by

_ o)k2[{\ - S){9 - <J2){\ - a2) + Sk2{l - a2){3 - a2)] , [8(1 - S) + 5Sk2]o)k2

P — A - T r s - , 777^ 777J777 TJTT-i 1 7T, n , n, _ n > (5-28)

y o)k 2

4a2[(í - S)a2 - Sk2(3 - a2)]

(1 - S + Sk2)(l + o2)2 4(1-S) + lSk

4(1-S + Sk2) '

(l-S + 4Sk2)a2

+

A = kc/(2(o), r2 = (ok(l- a2)/(2avg), (5.30)

r3=2co/ff, r4 = %aú2/a, r5 = (1 - a2)o)2/a2, r6 = 3(1 - a2)oú/a2. (5.31)

$( j ) = 2a>k cosh[2k(y + 1 )]/sinh2£, (5.32)

where a = tanh£. Note that /? diverges at (1 — S)a2 = Sk2(3 — a2), i.e., when the strictly inviscid

McGoldrick (1970) and Trulsen and Mei (1995, 1997) for nearly inviscid descriptions that ignore the mean flow.

5.2. Dynamics near the bicritical point

To examine the role of the mean flow near the bicritical point we first rescale the variables and parameters as in (2.10), namely

(x,t) = L(x,t), A±=L~1A±, (v, n) = L~\v, /t), (5.33)

and

xJ/m=L-2$m, f"=L-2fm, (5.34)

where \J/m = \J/m(x,y,T), fm = fm(x,x). Dropping the tildes and defining the bifurcation parameters p, and v, the damping rate parameter 8 and the slow time variable % as in (4.1) and (4.2), namely

¡i = n+L-\jl-2n2a), v = % + L~\v - 2n2a), 5 = L-3/25, x = L~1/2t, (5.35)

Eqs. (5.17)—(5.23) become

Af ^Af + in(A± -A*) = -L~1/2(Af + 5A±) + iL"1 [(/2 - 2n2a)ÁZf - (v - 2n2a)A±]

+ iL-1[(P\A±\2 +y\A^\2)A± + aAfx]

ro

i'yy, = ®l>?yyy i n " l < V < °> (5 3 7)

+ L~\±irx j g(y){il?)xdy + ir2{fn)x]A± + ---, (5.36)

^ = r,(\A-\2-\A+\2)x, ^y = r4(\A+\2-\A-\2) aty = 0, (5.38)

xjjm = 0, xJj™ = -r6[iA+A-e2lkLx + c.c. + \A-\2-\A+\2] aty = -í, (5.39)

A±(x + l,t,T)=A±(x,t,T), \¡jm{x+l,y,T) = \¡jm{x,y,T), (5.40)

cf. (4.3) and (4.4), where, because of the scaling (5.33), the spatial average (5.24) must be replaced by

rx+í

{G{x,y,t))x = {2f)~x G(z,y,t)dz, with 1/L4>?41. (5.41)

Jx—rf

The parameter s appearing in (5.37) is defined as

e=L3'2Cg/vg, (5.42)

by (5.20) and (5.23),

fm = -r5vg[\A+\2 + \A-\2- í (\A+\2 + \A-\2)dx]/(l-S), (5.43)

Jo

where the integral comes from volume conservation and we assumed

1 - 5 - 1 , (5.44) thereby excluding the capillary limit.

As in Section 4, we now write

A± = A± +L~1/2Af +L~xAf + ••• + {B±+L-xl2Bf+L-xBf + • • -)e±2mx + HOH, (5.45)

where

Ao=A+, B~ = -B+. (5.46)

Thus

\A+\2 - \A~\2 = 2A+B+e2mx + ce. + • • •, \A+\2 + \A~\2 = 2(\A+\2 + \B+\2) + •••. (5.47)

and so fm vanishes at leading order. Likewise, we suppose that

xjjm = x¡j{)(y,x)Q2mx + ^(j;,T)e2 i a x + c.c. + • • •, (5.48)

although \JA will not be needed in what follows. The slow function \J/o is found (upon substitution of (5.43), (5.47a) and (5.48) into (5.37)-(5.39)) to be given by

</Vyr = Slj/Oyyyy Ül - 1 < }> < 0, (5.49)

iAo = -2r3A+B+, ip0yy = 2r4A+B+ aty = 0, (5.50)

tA) = 0, xp()y = 2 Í V + 5 + at y = -1. (5.51)

Eqs. (4.9) and (4.10) continué to hold, and again yield no solvability condition. However, Eqs. (4.11) and (4.12) must be replaced by

m(Af -Af) = -dAf/dx - 8Af + i(/2 - 2n2a)Af - i(v - 2n2a)A0h + ip(\A±\2 + 2|5+|2>4+

+ iy[(\Af\2 + \Bf\2)A± +AfB*Bf] ± iA j g(y)^y dyB±, (5.52)

in(-Bf -Bf) = -dBf/dx - 5Bf + i(/2 - 2n2a)Bf - i(v + 2n2a)B± + ifi(\B±\2 + 2 | ^ |2) 5 ^

+ iy[{\Bf\2 + \Af\2)B± + BfAfAf] ± iA j g{y)^y dyAf,

(5.53) where \¡/f are defined as

Eqs. (5.52) and (5.53) are solvable only if the following conditions hold (cf. (4.13) and (4.14)): r0

A"+ 25A'+ d2A = 2n[fi-v +p(\A\2 + 2\B\2) + y\A\2]A + 2nr1 / g(y)(py dyB, (5.55)

B"+ 25B'+ 52B = 2n[fi + v-fi(\B\2+2\A\2)-y\B\2]B-2nr1 / g(y)<pydyA, (5.56)

where, as in Section 4,

A=A+=A~, B = B+ = -B~, q> = ^. (5.57)

Using these variables the mean flow equations (5.49)-(5.51) become

cpyyz = S(pyyyy in - 1 < y < 0, (5.58)

<p = -2r3AB, (pyy = 2r4AB aty = 0, (5.59)

<p = 0, (py = 2r6AB aty = -í. (5.60)

In the following we refer to (5.55), (5.6), (5.58)-(5.60) as the coupled amplitude-mean flow (CAMF) equations, and note that

• The CAMF equations are still invariant under the operations

A^y-A, A^elclA, (p^e~lci(p and B -»• -B, B -»• e1C2B, (p -»• el c> (5.61)

for all c\ and C2. Thus the spurious symmetry, noted in Section 4, is still present. Moreover, since the mean flow is not slaved to the surface waves it plays a dynamical role in the stability of such waves (see below), and henee in their dynamics.

• When \5\ ~ 1, as assumed above, the parameter s=dCg/(dvg), and henee (using (5.6) and (5.25a))

is (i) of the order of cj 4,1 if k is bounded above, and (ii) of the order of \¡k2 4,1 if k is large.

In fact, s is only logarithmically small (and so can be treated as 0(1)) if A: is logarithmically large compared to l/Cg.

• The CAMF equations show readily that the (leading order) mean flow is nonzero whenever AB ^ 0, Le., the mean flow vanishes only for the puré modes with m = 0 or 1.

5.3. Linear stability of the puré modes

The puré modes genérate no mean flow and henee are still given by (4.18), namely

\As\2 = ' - f i ^ ( 2 n \ \B2\ = (Ps = 0,

W2 = 9, = 0, \Bs\2 = ' + f l - ^( 2 n \ (5.62)

In the context of the Faraday system these solutions represent puré standing waves, with respective ly, N and N + 1 wavelengths in the period L. Fig. 4a shows the regions in the (S,k) plañe where the quantity /? + y is positive (unshaded) and negative (shaded). As in Section 4, we assume that

(a) 5

4h

3r k

1L

O 0.2 0.4 0.6 0.8 1

S

(b) 5

4

3

2

1

0

0 0.2 0.4 0.6 0.8 1

S

(c) 5

4

3 k

2

1

0

0 0.2 0.4 0.6 0.8 1

S

The results for the case /? + y < 0 follow from those below using the invariance of the CAMF equations under the operation

A^B, _-v,

_{P^-P, 7^-7, A}

_-r.

The first of the puré modes in (5.62) then bifurcates subcritically and henee is always unstable. The linear stability properties of the second puré mode in (5.62) are determined by replacing A by

BS(XXQX% + X2QXT), B by 5,(1 + YXQX% + Y2QXT) and q> by 2\BS\2(<PX2QX% + <PXXQX%) and linearizing

Eqs. (5.55), (5.56) and (5.58)-(5.60). There are then two types of potential instabilities. The puré mode instability satisfies

(X + 5?

2n

{$ + y)\Bs\2Yx +

fi-v + 2(p + y)\Bs Y1+(p + y)\Bs\2Y2 = 0,

(X + 5f

2% fi-v + 2(p + y)\Bs Y2 = 0,

(5.65)

(5.66)

withXi, X2 and q> slaved to Y\ and Y2. This system possesses a zero eigenvalue, with eigenfunction (Y\,Y2) = (iBs,—iBs), due to the invariance of Eqs. (5.55)-(5.60) with respect to the operation

(5.61b). The remaining three eigenvalues are strictly negative, as in Section 4. The mixed mode instability excites perturbations that are orthogonal to the mode in question. Thus X\ ^ 0, X2 ^ 0,

but Y\ = Y2 = 0. Both X\ and X2 satisfy identical equations: (X + 8f

2% fi + v-2p\Bs\2 -2r1\Bs d(y)#ydy

x = o,

where $ is the unique solution of X<Pyy = B'Pyyyy Ül - 1 < J/ < 0,

<P = - A , <P_{- A ,} yy = T4 at y = 0,

o, <z>

<Z> = 0, <Py = r6 at y = -\.

Integration of (5.68H5-70) for X > 0 yields

VX(y+l) VX(y + l)

<P = Ki sinh K7

c o

h ^ ± l > - l

+ r

{y+l),

(5.68)

(5.69)

(5.70)

(5.71)

where

K,

K?

( r3 + r6)coshTyX/s + r4(s/ X)[cosh T/X/S — 1] \JX~fs cosh \fJJs — sinh \JX~fs

( r3 + TÓ) sinh \fX~fs + r4(s//l)[sinh A/I/S — \fX¡¿\

while if X = 0

<P = r6(y+l)

'X/scosh wX/s — sinh v/A/s

r[6(r

+ r

) + r

](j + i)

+ ±[2(r

+ r

) + r

](y + i)

.

(5.72)

Note that all the eigenvalues X of (5.67) are doubled. These eigenvalues satisfy X2 + 25 X

2% + 2v - (3j8 + y)\Bs\2 - S)(X)\BS\2 = 0, where

(X) = 2rx / g(y)$ydy.

Thus, if X > 0,

°J{"X) 4k2 sinh 2k but when X = 0,

4k2

[(r6£), + KMX) - K2£)3(X)],

0(0):

sinh 2£ Here

sinh 2k 2k '

A>0i + ^[6(r

+ r

) + r

]0

+ ^P(r

+ r

) + r

]0

sinh(2£ + \/JJi) sinh(2£ — \/JJi) ^

i=^-L -\ i=^-L ~ 0 i

2{2k + Jlfz) 2(2k - •Jije)

(5.74)

(5.75)

(5.76)

(5.77)

cosh(2£ + v/I/s) — 1 cosh(2£ — v/I/s) — 1

2(2£ + y/JJz)

cosh 2k — 1 sinh 2k cosh 2k

+ .

4k2 " 3 4£3 2 F

One can check that, properly interpreted, the above expressions apply even when X < 0 or when it is complex. Note that /?, r3, r4 and TÓ depend only on k and 5, while 5 and s are determined by

the chosen valúes of Cg and L; fi and v are of course the two parameters that unfold the bicritical

point.

The characteristic equation (5.74) admits a stationary instability at

[3p + y + 2)(0)]\Bs\2/2, (5.78)

and the instability sets in as \BS\ increases if 3/? + y + 0 ( 0 ) > 0 and as \BS\ decreases if 3/? + y +

0 ( 0 ) < 0. The regions of positive (unshaded) and negative (shaded) valúes of 3/? + y + 0 ( 0 ) in the (S,k) plañe are shown in Fig. 4b. The instability is oscillatory with X = iXj, Xj > 0, if

15, SXi

1

4%

+

SXj[3p + y + S)R(Xi)]

(5.79)

710/(1/)' 4TC ' 2TC0/(/1/)

Fig. 5. The neutral steady and oscillatory instability curves of the second puré mode in (5.62), for 3=1, £ = 0.1 and (a)

k = 0.5, S = 0,(b)k = 0.5, S = 0.8, (c) k = 2, S = 0, and (d) k = 1, S = 0.8. The puré mode is stable only below these

curves. For comparison, the neutral instability curve without mean flow (i.e., £^(0) = 0) is shown using a dashed line. In all cases the results correspond to the bifurcation diagram in Fig. 2(a); to obtain this diagram for cases (b) and (d) one must first apply the transformation (5.64).

región in the (\Bs\2,v) plañe where the puré mode (0,BS) is stable (unshaded) and unstable (shaded).

Note that for sufficiently large detuning the presence of the Hopf bifurcation reduces dramatically the región of stability of the puré mode, especially in cases (a,c). The reduction is smaller in cases (b,d); since /? + y < 0 in these plots the transformation (5.64) has been applied.

5.4. Linear stability of the mixed modes

The mixed mode solutions of the CAMF equations (5.55), (5.56), and (5.58)-(5.60) are given by

(\M\\B]\) =

(2nfi - d

, -2nfi + S

)

₊

2n[p-y + ®(0)] ' 30 + y + 0(O)'

bifürcation points at the end of the mixed mode branch are unchanged, although this is not true of the secondary Hopf bifürcation (see below).

The linear stability properties of the mixed modes are determined by replacing A, B, and q> by

AS(\+XXQXX+X2QXX\ BS(\ + YXQXX + Y2QXX\ and (ps+ÁsBs(<¡>XQXx + <¡>2QXX), respectively, and linearizing

Eqs. (5.55), (5.56) and (5.58)-(5.60). From the latter it follows that: fa = 2(X2 + Y1 )<P(X, y), <j)2= 2{XX + Y2 )<P(X, y),

while the former show that (X\,X2,Y\,Y2) satisfy the algebraic equation

'{X +

5f

(5.81)

271 fi + v - 2(3 + y)\As\2 -2p\Bs X-(p + y)\As\2X2

[2j8 + ^ ( O ) ] ^2^ - 2p\B3\2Y2 - ^(X)\BS\2(X1 + Y2) = 0, (5.82)

together with three equations obtained by (i) interchanging X\ <-> X2 and Y\ <-> Y2, (ii) interchanging As <-> Bs, X\ <-> Y\, X2 <-> Y2, and changing the signs of (fi,y,v,!3), and (iii) interchanging X\ ^ X2

and Y] <-+ Y2 in the result of (ii). Here 3){X) is as defined in (5.76) and (5.77). Using (5.80) it

follows that X+ =X1+X2 and Y+ = Y1 + Y2 satisfy X2 + 2SX

2% 2(p + y)\As\2-(®(X)-®(0))\Bs X+ - [4p + Si{Q) + Si(X)]\Bs\2YJ'

[4/3 + 3i(0) + 3i(X)]\As\2X++ X2 + 2SX

2% + 2(3 + y)\Bs\2 + (mX)- mO))\As Y'

while X~ =XX-X2 and Y 'X2 + 25X

Y\ — Y2 satisfy

2n (2(X) - ^ ( 0 ) ) | 5 , x~ + [Si(X) - ^ ( o ) ] ^ !2^ - = o,

[Si(X)-Si{Q)]\As\2X- + X

2 + 25X

2% + (®(X) - ®(0))\AS Y~ 0.

:0,

(5.83)

:0,

(5.84)

(5.85)

(5.86)

We may identify the former equations as describing instability with respect to amplitude or standing wave perturbations (the spatial phase of the mixed mode remains fixed by these perturbations), while the latter describe instability with respect to phase perturbations.

D(X) — D(0) = aX + 0(X2), and the system (5.85) and (5.86) reduces to a characteristic equation of

the form

X2 a 27icr(/2 - 52/2n)

s

~

n-j +

w)

0, (5.87)

indicating the presence of a parity-breaking bifurcation at

fí = fip = l- + J-[P-y+D(0)]. (5.88)

This instabihty is only possible because the coupling to the mean flow results in a transcendental characteristic equation with a larger number of zero eigenvalues, and it produces traveling waves that drift steadily either to the left or the right. In addition we may have purely imaginary roots of the characteristic equation. This possibility is also a consequence of the coupling to the mean flow, and it generates the so-called direction-reversing waves, Le., an oscillation in the spatial phase of the mixed mode (Landsberg and Knobloch, 1991). We do not pursue here these instabilities further.

6. Concluding remarks

In this paper we have examined the dynamics of the nearly inviscid Faraday system near codimension-two points where the neutral stability curves for adjacent modes cross. As is well-known such points provide the key to the nonlinear phenomena associated with the transition from one mode to another, since near such points the necessary secondary bifurcations all occur at small amplitude and so are (usually) analytically accessible. We focused on large aspect ratio domains which permit the presence of large-scale mean flow, and explored the role played by this flow in the resulting transition. In the absence of a mean flow we showed that the basic equations reduce to equations already familiar from the theory of nonresonant interaction of distinct modes. The different possibil-ities are summarized in Fig. 2. An important feature of these diagrams is that one of the two modes is necessarily subcritical while the other is supercritical. This property is a consequence of the fact that the primary Faraday resonance in an inviscid fluid produces an instabihty whose direction of branching depends on the sign of the detuning (relative to optimal).

indicated by shading. In cases (a,c) the results of Fig. 4 show that /? + y > 0, 3/? + y + 0 ( 0 ) > 0, p - y + 0 ( 0 ) > 0, while in cases (b,d) p + y < 0, 3j3 + y + 0 ( 0 ) < 0, jg - y + 0 ( 0 ) < 0. The transformation (5.64) shows that the bifürcation diagrams in both cases are topologically the same provided one exchanges Bs for As and changes the sign of v. This diagram is shown in Fig. 2a, and

shows that the supercritical mode bifurcates first and is initially stable. This mode loses stability at finite amplitude to a mixed mode which inherits the stability before undergoing a Hopf bifürcation. It is the first of these bifurcations that may be preceded by the bifürcation to direction-reversing waves. Fig. 5 also shows the presence of an interesting codimension-two point, at which the steady and Hopf bifurcations on the supercritical puré mode branch come in simultaneously. This is an interaction between a pitchfork and a symmetry-breaking Hopf bifürcation, and such interactions may lead to a variety of new dynamical phenomena, cf. Landsberg and Knobloch (1993).

Similar computations for the stability of the mixed modes in the presence of mean flow show that the mixed modes can lose stability in two ways, either via a parity-breaking steady-state bifürcation producing traveling waves, or via a Hopf bifürcation. Of the latter there are two types, a Hopf bifürcation in which the oscillations respect the spatial phase of the mixed modes (producing so-called standing waves) and a Hopf bifürcation which produces oscillations in the spatial phase (and so results in direction-reversing waves). Only the standing oscillations are possible in the absence of the mean flow (cf. Fig. 2a) and these must be present even with mean flow in diagrams such as Fig. 2a where (in the standing wave subspace) the stability assignments at the two ends of the mixed mode branch differ. In particular, in Fig. 5a the standing wave instability will necessarily destabilize the mixed modes in the región to the left of the codimension-two point. Although we have not pursued the mixed mode instabilities further it is clear that these permit the presence of new types of higher codimension points in parameter space (for example, where the two Hopf bifurcations coincide), and henee new types of rich dynamics.

Acknowledgements

This work was supported in part by the NASA Microgravity Sciences Program under Grant NAG3-2152 and by the National Science Foundation under Grant DMS-0072444. We are grateful to Dr. M. Higuera for providing Figs. 4 and 5.

References

Christodoulides, P., Dias, F., 1994. Resonant gravity-capillary interfacial waves. J. Fluid Mech. 265, 303-343.

Ciliberto, S., Gollub, J.P., 1985. Chaotic mode competition in parametrically forced surface waves. J. Fluid Mech. 158, 381-398.

Craik, A.D.D., 1985. Wave Interactions and Fluid Flows. Cambridge University Press, Cambridge.

Crawford, J.D., Knobloch, E., Riecke, EL, 1990. Period-doubling mode interactions with circular symmetry. Physica D 44, 340-396.

Cross, M.C., Hohenberg, P.C., 1993. Pattern formation outside of equilibrium. Rev. Mod. Phys. 65, 851-1112.

Davey, A., Stewartson, S., 1974. On three-dimensional packets of surface waves. Proc. R. Soc. London A 338, 101-110. Decent, S.P., Craik, A.D.D., 1999. Sideband instability and modulations of Faraday waves. Wave Motion 30, 43-55. Douady, S., Fauve, S., Thual, O., 1989. Oscillatory phase modulation of parametrically forced surface waves. Europhys.

Faraday, M., 1831. On the forms and states assumed by fluids in contact with vibrating elastic surfaces. Phil. Trans. R. Soc. London 121, 319-340.

Fauve, S., 1995. Parametric instabilities. In: Martmez-Mekler, G., Seligman, T.H. (Eds.), Dynamics oí Nonlinear and Disordered Systems. World Scientific, Singapore, pp. 67-115.

Higuera, M., Vega, J.M., Knobloch, E., 2001. Interaction oí nearly-inviscid, multi-mode Faraday waves and mean flows. In: Bonilla, L.L., Platero, G., Reguera, D., Rubi, J.M. (Eds.), Coherent Structures in Complex Systems. Springer, Berlin, pp. 328-337.

Higuera, M., Nicolás, J.A., Vega, J.M., 2002a. Weakly nonlinear non-axisymmetric oscillations oí capillary bridges at small viscosity. Phys. Fluids 14, 3251-3271.

Higuera, M., Porter, J., Knobloch, E., 2002b. Heteroclinic dynamics in the nonlocal parametrically driven nonlinear Schródinger equation. Physica D 162, 155-187.

Higuera, M., Vega, J.M., Knobloch, E., 2002c. Coupled amplitude-mean flow equations for nearly-inviscid Faraday waves in modérate aspect ratio containers. J. Nonlinear Sci. 12, 505-551.

Iskandarani, M., Liu, P.L.F., 1991. Mass transport in three-dimensional water waves. J. Fluid Mech. 231, 417-437. Jones, M.C.W., 1992. Nonlinear stability oí resonant capillary-gravity waves. Wave Motion 15, 267-283.

Knobloch, E., Martel, C , Vega, J.M., 2002. Coupled mean flow-amplitude equations for nearly inviscid parametrically driven surface waves. Ann. N.Y. Acad. Sci. 974, 201-219.

Knobloch, E., Vega, J.M., 2002. Nearly inviscid Faraday waves. In: Newton, P., Holmes, P., Weinstein, A. (Eds.), Geometry, Mechanics and Dynamics: Volume in Honor oí the 60th Birthday oí J.E. Marsden. Springer, Berlin, pp. 181-222.

Landsberg, A.S., Knobloch, E., 1991. Direction-reversing traveling waves. Phys. Lett. A 159, 17-20.

Landsberg, A.S., Knobloch, E., 1993. New types oí waves in systems with 0(2) symmetry. Phys. Lett. A 179, 316-324. Lapuerta, V., Martel, C , Vega, J.M., 2002. Weakly dissipative Faraday waves in 2D large aspect ratio annuli. Physica D

173, 178-203.

Liu, A.K., Davis, S.H., 1977. Viscous attenuation oí mean drift in water waves. J. Fluid Mech. 81, 63-84. Longuet-Higgins, M.S., 1953. Mass transport in water waves. Phil. Trans. R. Soc. A 245, 535-581.

Martel, C , Knobloch, E., Vega, J.M., 2000. Dynamics oí counterpropagating waves in parametrically forced systems. Physica D 137, 94-123.

Martel, C , Vega, J.M., Knobloch, E., 2003. Dynamics oí counterpropagating waves in parametrically driven systems: dispersión vs. advection. Physica D 174, 198-217.

Martín, E., Martel, C , Vega, J.M., 2002. Drift instability of standing Faraday waves in an annular container. J. Fluid Mech. 467, 57-79.

McGoldrick, L.F., 1970. On Wilton ripples: a special case of resonant interactions. J. Fluid Mech. 42, 193-200. Miles, J., Henderson, D., 1990. Parametrically forced surface waves. Annu. Rev. Fluid Mech. 22, 143-165.

Pierce, R.D., Knobloch, E., 1994. On the modulational stability of traveling and standing water waves. Phys. Fluids 6, 1177-1190.

Rayleigh, Lord, 1883. On the circulation of air observed in Kundt's tubes, and on some allied acoustical problems. Phil. Trans. R. Soc. London. 175, 1-21.

Rayleigh, Lord, 1945. The Theory of Sound, Vol. II. Dover, New York. Riley, N., 2001. Steady streaming. Annu. Rev. Fluid Mech. 33, 43-65.

Schlichting, H., 1932. Berechnung ebener periodischer Grenzschichtstrómungen. Phys. Z. 33, 327-335. Schlichting, H., 1968. Boundary Layer Theory. McGraw-Hill, New York.

Simonelli, F., Gollub, J.P., 1989. Surface wave mode interactions: effects of symmetry and degeneracy. J. Fluid Mech. 199, 471-494.

Trulsen, K., Mei, C.C., 1995. Modulation of three resonating gravity-capillary waves by a long gravity wave. J. Fluid Mech. 290, 345-376.

Trulsen, K., Mei, C.C., 1997. Effect of weak wind and damping on Wilton's ripples. J. Fluid Mech. 335, 141-163. Umeki, M., 1991. Faraday resonance in rectangular geometry. J. Fluid Mech. 227, 161-192.